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Properties

Label 16.0.21662596979...0625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{8}\cdot 29^{4}\cdot 941^{4}$
Root discriminant $28.74$
Ramified primes $5, 29, 941$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T969

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![41, -401, 1646, -3314, 4216, -4622, 4638, -3842, 2832, -1904, 1142, -566, 246, -98, 34, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 34*x^14 - 98*x^13 + 246*x^12 - 566*x^11 + 1142*x^10 - 1904*x^9 + 2832*x^8 - 3842*x^7 + 4638*x^6 - 4622*x^5 + 4216*x^4 - 3314*x^3 + 1646*x^2 - 401*x + 41)
 
gp: K = bnfinit(x^16 - 8*x^15 + 34*x^14 - 98*x^13 + 246*x^12 - 566*x^11 + 1142*x^10 - 1904*x^9 + 2832*x^8 - 3842*x^7 + 4638*x^6 - 4622*x^5 + 4216*x^4 - 3314*x^3 + 1646*x^2 - 401*x + 41, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 34 x^{14} - 98 x^{13} + 246 x^{12} - 566 x^{11} + 1142 x^{10} - 1904 x^{9} + 2832 x^{8} - 3842 x^{7} + 4638 x^{6} - 4622 x^{5} + 4216 x^{4} - 3314 x^{3} + 1646 x^{2} - 401 x + 41 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(216625969799691187890625=5^{8}\cdot 29^{4}\cdot 941^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.74$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $5, 29, 941$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5} a^{8} + \frac{1}{5} a^{7} - \frac{1}{5} a^{6} + \frac{2}{5} a^{5} - \frac{2}{5} a^{4} + \frac{1}{5} a^{3} + \frac{1}{5} a^{2} + \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{9} - \frac{2}{5} a^{7} - \frac{2}{5} a^{6} + \frac{1}{5} a^{5} - \frac{2}{5} a^{4} + \frac{1}{5} a^{2} - \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{5} a^{10} - \frac{1}{5} a^{6} + \frac{2}{5} a^{5} + \frac{1}{5} a^{4} - \frac{2}{5} a^{3} + \frac{1}{5} a^{2} - \frac{2}{5} a + \frac{2}{5}$, $\frac{1}{5} a^{11} - \frac{1}{5} a^{7} + \frac{2}{5} a^{6} + \frac{1}{5} a^{5} - \frac{2}{5} a^{4} + \frac{1}{5} a^{3} - \frac{2}{5} a^{2} + \frac{2}{5} a$, $\frac{1}{5} a^{12} - \frac{2}{5} a^{7} - \frac{1}{5} a^{4} - \frac{1}{5} a^{3} - \frac{2}{5} a^{2} + \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{13} + \frac{2}{5} a^{7} - \frac{2}{5} a^{6} - \frac{2}{5} a^{5} - \frac{1}{5} a^{2} + \frac{2}{5}$, $\frac{1}{1987435} a^{14} - \frac{7}{1987435} a^{13} - \frac{172453}{1987435} a^{12} - \frac{157652}{1987435} a^{11} + \frac{186337}{1987435} a^{10} - \frac{82939}{1987435} a^{9} + \frac{118802}{1987435} a^{8} + \frac{665}{4789} a^{7} + \frac{162184}{1987435} a^{6} - \frac{941408}{1987435} a^{5} - \frac{887978}{1987435} a^{4} + \frac{912201}{1987435} a^{3} - \frac{145294}{397487} a^{2} - \frac{674028}{1987435} a + \frac{648721}{1987435}$, $\frac{1}{1393191935} a^{15} + \frac{343}{1393191935} a^{14} - \frac{121408438}{1393191935} a^{13} + \frac{32098269}{1393191935} a^{12} + \frac{73396438}{1393191935} a^{11} - \frac{58085959}{1393191935} a^{10} + \frac{127302543}{1393191935} a^{9} + \frac{118571666}{1393191935} a^{8} - \frac{24531878}{278638387} a^{7} + \frac{630986681}{1393191935} a^{6} - \frac{313686324}{1393191935} a^{5} + \frac{675490174}{1393191935} a^{4} - \frac{598458629}{1393191935} a^{3} + \frac{70166332}{278638387} a^{2} - \frac{350532309}{1393191935} a + \frac{144772541}{1393191935}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}$, which has order $2$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $7$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 36913.3751226 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T969:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 44 conjugacy class representatives for t16n969
Character table for t16n969 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.23525.1, 8.0.16049343125.1, 8.8.16049343125.1, 8.0.465430950625.2

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 16 siblings: data not computed
Degree 32 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$29$29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
941Data not computed